330 LL VELOCITIES. RIGID BOOT. 



changes made in the co-ordinates ar, y, zof a particle, wo have 



wlm-h not on the ri^id body are su]] 

 to keep it in equilibrium, we have by 



:.v=o, 2r=o, 



r,~) = 0, 



Multiply the first of these equations by a, the 



the third by <\ and the sixth by #, and add; we then find 



or 1 (A'&c + r% + Z8z) = 0. 



Let P l denote the force of which JT t , Y lt Z are the 

 noiu-iiTs, and P 2 , P 8 , ...... liavc similar meanings; and h-t 



f/> a . ...... be the resolved virtual velocities eo 



ing to these forces; then, by Art. 250, the above 

 may be written 



This proves the principle in the case of a ri 



Conversely, if the sum of the products of the forces and Un- 

 resolved virtual velocities vanishes for every possible disp 

 mcnt of a rigid body, the forces keep the body in equilibrium. 



For suppose, in the first plare. the body is so disj.l 

 that every point of it moves parallel to the axis of x u 

 space a ; then we have, by hypothesis, 



SJa-0; 



fore ^ A' = 0. 



Similarly, by suitable displacements, we may prove that 

 2F=0, and 



Next, suppose the body turned round the axis of z through 

 a small angle 6 ; then, by hypothesis, 



